Unformatted text preview: X i are independent Bernoulliâ€™s. 3. Poisson: E e tX = X e tk eÎ» Î» k k ! = eÎ» X ( Î»e t ) k k ! = eÎ» e Î»e t = e Î» ( e t1) . 4. Exponential: E e tX = Z âˆž e tx Î»eÎ»x dx = Î» Î»t if t < Î» and âˆž if t â‰¥ Î» . 5. N (0 , 1): 1 âˆš 2 Ï€ Z e tx ex 2 / 2 dx = e t 2 / 2 1 âˆš 2 Ï€ Z e( xt ) 2 / 2 dx = e t 2 / 2 . 6. N ( Î¼,Ïƒ 2 ): Write X = Î¼ + ÏƒZ . Then E e tX = E e tÎ¼ e tÏƒZ = e tÎ¼ e ( tÏƒ ) 2 / 2 = e tÎ¼ + t 2 Ïƒ 2 / 2 . Proposition 12.1. If X and Y are independent, then m X + Y ( t ) = m X ( t ) m Y ( t ) . Proof. By independence and Proposition 11.1, m X + Y ( t ) = E e tX e tY = E e tX E e tY = m X ( t ) m Y ( t ) . 24...
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 Spring '09
 wen
 Bernoulli, Binomial, Normal Distribution, Variance, Probability theory, moment generating function, ETX, Var X1 /n

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